The Mathematical Education (한국수학교육학회지시리즈A:수학교육)

Korean Society of Mathematical Education (한국수학교육학회)
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Elementary school students' metaphors of angle concepts

초등학생의 각 개념 형성에 나타난 수학적 은유

  • Kim Sangmee (Chuncheon National University of Education)
  • Received : 2023.02.09
  • Accepted : 2023.02.13
  • Published : 2023.02.28
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Abstract

This study used metaphors as a analysis tool to investigate elementary school students' formation and development of angle concepts. For this purpose, the students were asked to write words associated with angle, right angle, acute angle and obtuse angle and to explain why. In case of angle and right angle, responses of 268 students from 3rd to 6th graders were analyzed and for acute angle and obtuse angle, those of 192 students from 4th to 6th graders were examined. As the results of categorizing the metaphors, they can be classified into categories such as; (1) qualitative aspects: 'things metaphor', 'personality metaphor', 'emotions metaphor' etc., (2) quantitative aspects: 'motions metaphor', 'changes metaphor', 'emotions metaphor' etc., and (3) relational aspects: 'shape relations metaphor.' The metaphoric expressions were prominent in 'qualitative aspects' associated with shapes. As for the other aspects, 'quantitative aspect'- the size of angles and the amount of spread and 'relational aspects' - elements of angle and relationship with another shapes, the frequency increses were shown to as grade levels were up. In case of right angle and acute angle, 'qualitative aspects' associated with shapes were outstanding and the frequency of the metaphoric expressions of obtuse angle was distributed similarly in three aspects. As the figure strand and the measurement strand are integrated to an strand in the 2022 revised curriculum, we need more discussion of multifaced aspects of angle and the learning sequences in the 'figure and measurement' strand.

이 연구는 초등학생이 각의 다면성을 어떻게 형성하고 학년이 올라가면서 초등학생의 각 개념은 어떻게 변화하는가를 은유 분석하였다. 초등학교 각 개념 학습 요소인 각, 직각, 예각, 둔각에 대하여, 이 용어를 생각하면 떠오르는 것을 낱말로 표현하고 그 근거를 서술하도록 하였다. 각과 직각은 3학년 1학기에 학습하므로 3~6학년 총 268명의 응답을 분석 대상으로 하였고, 예각과 둔각은 4학년 1학기에 학습하므로, 4~6학년 총 192명의 응답을 분석 대상으로 설정하였다. '은유적 표현'과 그 '근거'를 짝지어 은유적 표현을 정리하고, 기하적 도형이라는 질적 측면, 측정 및 회전량이라는 양적 측면, 점과 선의 구성 요소와의 관계적 측면에서 코드화하였다. 은유적 표현을 범주화한 결과, 질적 측면에서 <사물의 은유>, <인간형의 은유>, <감정의 은유> 범주 등, 양적 측면에서 <움직임의 은유>, <변화의 은유>, <감정의 은유> 범주 등, 관계적 측면에서 <도형 관계의 은유> 범주를 찾았다. 초등학생의 은유적 표현은 모양으로 접근하는 각의 질적 측면에서 가장 많이 나타났고, 학년이 올라가면서 각의 크기 및 벌어진 정도의 양적 측면이나 각의 구성 요소 및 다른 도형과의 관계적 측면이 증가하였다. 직각과 예각은 모양의 접근이 두드러졌고 둔각은 세 가지 접근의 빈도 분포가 유사하였다. 이 연구에서 추출한 초등학생의 은유적 표현은 각 개념 형성을 파악하는 기초 자료로 활용되거나 수업 구성 및 학습 자료로 활용될 수 있을 것이다. 다면적인 각 개념의 형성을 위하여 차시별 도입 방법만이 아니라 관련 학습 내용 간의 학습 계열의 추가적인 논의가 필요하고, 2022 개정 수학과 교육과정에서 도형과 측정 영역이 하나의 영역으로 변경되면서 각의 다면성과 연계하여 학습 계열 설정의 논의가 더욱 중요한 시기이다.

Keywords

Acknowledgement

This work was supported by Chuncheon National University of Education Grant in 2021.

References

  1. Armstrong, S. L. (2008). Using metaphor analysis to uncover learners' conceptualizations of academic literacies in post secondary developmental contexts. The International Journal of Learning, 15(9), 211-218. https://doi.org/10.18848/1447-9494/CGP/v15i09/45948
  2. Bullough, R.V. (1991). Exploring personal teaching metaphors in preservice teacher education. Journal of Teacher Education, 42(1), 43-51. https://doi.org/10.1177/002248719104200107
  3. Cameron, L. (2003). Metaphor in educational discourse. Continuum.
  4. Cassel, D., & Vincent, D. (2011). Metaphors reveal preservice elementary teachers' views of mathematics and science teaching. School Science and Mathematics, 111(7), 319-324. https://doi.org/10.1111/j.1949-8594.2011.00094.x
  5. Chapman, O. (1997). Metaphor in the teaching of mathematical problem solving. Educational Studies in Mathematics, 32(3), 201-228. https://doi.org/10.1023/A:1002991718392
  6. Clandinin, D. J. (1986). Classroom practice: Teacher images in action. The Falmer Press.
  7. Chiu, M. M. (1994). Metaphorical reasoning in mathematics: Experts and novices solving negative number problems. Paper Presented at the Annual Meeting of the American Education Association, April, 4-8. LA: New Orleans (ERIC Document Reproduction Service No. ED 374-988).
  8. Crompton, H. (2015). Understanding angle and angle measure: A design-based research study using context aware ubiquitous learning. International Journal forTechnology in Mathematics Education, 22(1), 19-30. doi: 10.1564/tme_v22.1.02
  9. Danesi, M (2007). A conceptual metaphor framework for the teaching of mathematics. Studies in Philosophy and Education, 26, 225-236. https://doi.org/10.1007/s11217-007-9035-5
  10. Devichi, C., & Munier, V. (2013). About the concept of angle in elementary school: Misconceptions and teaching sequences. The Journal of Mathematical Behavior, 32(1), 1-19. https://doi.org/10.1016/j.jmathb.2012.10.001
  11. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Reidel Publishing Company.
  12. Fyhn, A. B. (2006). A climbing girl's reflections about angles. The Journal of Mathematical Behavior, 25(2), 91-102. https://doi.org/10.1016/j.jmathb.2006.02.004
  13. Henderson, D. W. & Taimina, D. (2007). Experiencing meanings in geometry. In D. Pimm, M. Sinclair, & W. Higginson (Eds.), Mathematics and the Aesthetic (pp. 58-83). Springer-Verlag.
  14. Heaton, R. M. (2000). Teaching mathematics to the new standards: Relearning the dance. The Teachers College Press.
  15. Johnson, M. (1981). Introduction: Metaphor in the philosophical tradition. In M. Johnson (Ed), Philosophical perspectives on metaphor (pp. 3-47). The University of Minnesota Press.
  16. Ju, M. K. & Kwon, O.N. (2003). Students' conceptual metaphor of differential equations: A sociocultural perspective on the duality of the students' conceptual model. School Mathematics, 5(1), 135-149
  17. Keiser, J. (2004). Struggles with developing the concept of angle: Comparing 6th grade students' discourse to the history of the angle concept. Mathematical Thinking and Learning, 6(3), 285-306. http://dx.doi.org/10.1207/s15327833mtl0603_2
  18. Kim, D. & Kim, M. K. (2020). A study on understanding and misconception about angle of Elementary 4th Grade Students. School Mathematics, 22(2), 183-203. https://doi.org/10.29275/sm.2020.06.22.2.183
  19. Kim, S. (2005). Metaphors on mathematics teaching. School Mathematics, 7(4), 445-467.
  20. Kim, S. & Shin, I. (2007). On the Mathematical Metaphors in the Mathematics Classroom. Education of Primary School Mathematics, 10(1), 29-39.
  21. Kim, S. (2018a). Angle concepts and introduction methods of angles in elementary mathematics textbooks. Education of Primary School Mathematics, 21(2), 209-221. https://doi.org/10.7468/jksmec.2018.21.2.209
  22. Kim, S. (2018b). Research on the definitions of angle in the past Korean elementary mathematics textbooks. Journal of Educational Research in Mathematics, 28(3), 265-282. https://doi.org/10.29275/jerm.2018.08.28.3.265
  23. Kim, S. & Heo, H. (2022). The concept of the angle presented in the middle school mathematics textbooks. The Mathematical Education, 61(2), 305-322. https://doi.org/10.7468/MATHEDU.2022.61.2.305
  24. Kovecses, Z. (2010). Metaphor: A practical introduction (2nd ed). Oxford University Press.
  25. Kovecses, Z. (2015). Where metaphors come from: Reconsidering context in metaphor. Oxford University Press.
  26. Krussel, L., Edward, B., & Springer, G. T. (2004). The teacher's discourse moves: A framework for analyzing discourse in mathematics classrooms. School Science and Mathematics, 104(7), 307-312. https://doi.org/10.1111/j.1949-8594.2004.tb18249.x
  27. Lakoff, G. (1987). Women, fire, and dangerous things: What categories reveal about the mind. University of Chicago Press.
  28. Lakoff, G. (1993). The contemporary theory of metaphor. In A. Ortony (Ed.), Metaphor and thought (2nd ed.) (pp. 202-251). Cambridge University Press
  29. Lakoff, G., & Johnson, M. (1980). Metaphors we live by. Chicago University Press.
  30. Lakoff, G., & Johnson, M. (1999). Philosophy in the flesh: The embodied mind and its challenge to Western thought. Basic Books.
  31. Lakoff, G., & Nunz, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.
  32. Lee, C. H. (2001). An analysis on angle concepts in mathematics education. Journal of the Korea Society of Educational Studies in Mathematics: School Mathematics, 3(1), 25-44.
  33. Ministry of Education (2015). Mathematics curriculum. Ministry of Education Notice 2015-74 [supplement 8].
  34. Ministry of Education (2022). Mathematics curriculum. Ministry of Education Notice 2022-33 [supplement 8].
  35. Mitchelmore, M. C. (1998). Young students' concepts of turning and angle. Cognition and Instruction, 16(3), 265-284. https://doi.org/10.1207/s1532690xci1603_2
  36. Mitchelmore, M. C., & White, P. (2000). Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics, 41(3), 209-238. https://doi.org/10.1023/A:1003927811079
  37. Neyland, J. (2003). Playing outside: An introduction to jazz metaphor in mathematics education. Australian Senior Mathematics Journal, 18(2), 8-18.
  38. Nolder, R. (1991). Mixing metaphor and mathematics in the secondary classroom. In K. Durkin & B. Shire (Eds.), Language in mathematical education: Research and practice (pp. 561-578). Open University Press.
  39. Presmeg N. C. (1997). Reasoning with metaphors and metonymies in mathematics learning. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 267-279). Lawrence Erlbaum Associates.
  40. Reddy, M. (1993). The conduit metaphor: A case of frame conflict in our language about language. In A. Ortony (Ed.), Metaphor and thought (2nd ed.) (pp. 164-201). Cambridge University Press.
  41. Schmitt, R. (2005). Systematic metaphor analysis as a method of qualitative research. The Qualitative Report, 10(2), 358-394. https://doi.org/10.46743/2160-3715/2005.1854
  42. Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1), 44-54.
  43. Sfard, A.(1997) Commentary: On the metaphorical roots of conceptual growth. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 339-371). Lawrence Erlbaum Associates.
  44. Sfard, A.(1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4-13. https://doi.org/10.3102/0013189X027002004
  45. Sfard, A.(2000). Steering (dis)course between metaphor and rigor: Using focal analysis to investigate and emergence of mathematical objects. Journal for Research in Mathematics Education, 31(3), 296-327. https://doi.org/10.2307/749809
  46. Sticht, T. G. (1993). Educational uses of metaphor. In A. Ortony (Ed.), Metaphor and thought (2nd ed.) (pp. 621-632). Cambridge University Press.
  47. Park, C. K. (2017). Metaphors for Mathematics and Philosophical Problems. Journal for History of Mathematics, 30(4), 247-258. https://doi.org/10.14477/JHM.2017.30.4.247
  48. Reeder, S., Utley, J., & Cassel, D. (2009). Using metaphors as a tool for examining preservice elementary teachers' beliefs about mathematics teaching and learning. School Science and Mathematics, 109(5), 290-297. https://doi.org/10.1111/j.1949-8594.2009.tb18093.x